ON THE NEW APPROXIMATION TO NON-CENTRAL i-distributions. Dedicated to Professor Nariaki Sugiura on his 60th birthday

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2 J. Japan Statist. Soc. Vol. 25 No ON THE NEW APPROXIMATION TO NON-CENTRAL i-distributions Dedicated to Professor Nariaki Sugiura on his 60th birthday Masafumi Akahira*, Michikazu Sato* and Norio Torigoe* Recently a new approximation to a percentage point of non-central t-distributions was proposed by Akahira [1]. In this paper the approximation formula is presented and the existence and uniqueness of a solution of the equation on the formula is proved. Numerical results are also given. 1. Introduction Suppose that Xl,..., Xn 1 and Yl,..., Yn 2 are random samples from normal distributions with means Jll and #2, respectively, and a common variance (J"2. Letting we define the statistic T by T: X-y Then it is well known that T has a non-central t-distribution t (v, 0) with v degrees of freedom and a non-centrality parameter 0, where v: =nl+n2-2, 0: =~nln2/(nl+n2)d and D: = (#1-#2)/(J" and that T plays an important part in the testing problem on difference between #1 and #2, which also derives a confidence interval for it. In order to consider such an inference we need a percentage point of the non-central t-distribution. It is quite difficult to obtain the value analytically, however, since the density has the following form:._ /2 (.J20)k r[(v+k+1)/2j(_t_)k( ~)-CIJ+k+I)/2 f T (t, v, 0).- 2J e k, / r( /2) /- 1 + Ic=O 'V' TTV V 'V' V V for - 00 <t< 00. (For tables on percentage points of non-central t-distributions, see, e.g. Bagui [2J, IMS [4J and Yamauti et al. [15J.) Hence, we will consider approximation formulae for a percentage point of non-central t-distributions (Owen [llj and Johnson and Kotz [6J). In this Received November, Revised February, Accepted March, * Institute of Mathematics, University of Tsukuba, Ibaraki 305, Japan. This research was supported in part by Grant-in-Aid for Science Research No , Ministry of Education, Science and Culture, Japan

3 2 J. J AP AN STATIST. SOC. Vol. 25 No paper, we discuss the approximation formulae of Jennett and \Velch [5J, Johnson and Welch [7J, and van Eeden [13J, which are well known among others. We also present a new higher order approximation formula developed by Akahira [IJ, which is derived from the Cornish-Fisher expansion for the statistic based on a linear combination of a normal random variable and a chirandom variable. Akahira [1 J illustrated that this new approximation formula is numerically better than the others. In addition, we discuss the existence and uniqueness of a solution of the equation on the new approximation f0f111ula since it has an implicit form. More detailed numerical comparison of the formulae above are made and tables on values of percentage points computed from the new approximation formula are given. 2. Approximations to a percentage point of non-central t-distribution Suppose that X is a normal random variable with mean 0 and variance 1, vs 2 is a random variable according to a chi-square distribution with v degrees of freedom, and X and vs 2 are independent. Denoting Tv.;;: =X/..JS 2, we see that Tv,Q has a non-central t-distribution t(v, 0). Letting 5: =..J 52, we have Tv,Q =X/S. Among approximation formulae for a percentage point of the non-central t-distribution, we present some well-known formulae. Since P{Tv.Q~t}=P{X -ts~o}, we see that 5 is asymptotically normally distributed with mean b v : = E[SJ =..J2/vr[(v+l)/2J/r(v/2), and variance V[SJ=I-b~. When v is large, X-tS is asymptotically normally distributed with mean o-tb ll and variance 1+ t2(i-b~). For any a with O<a<l, we have hence a=p{t",>t"}=cl-<pcl:~;~~b;)), (2.1) t == obv+ua..jb~+(i-b~)(02-u;) (Jennett and Welch [5J) a. b~-u;(i-b~) where C/J(x) is the standard nornlal distribution function and U a is the upper 100a percentile of the standard normal distribution. Letting b v ~ 1 and 1-b~ ~ 1/2v in (2.1), we get (2.2) t == 0 +ua..ji + (02-u;)/(2v) (Johnson and Welch [7J). a. 1-u;/(2v) (Masuyama [10J obtained values of this approximation using an improved binomial probability paper.) Now we will note the approximation formula (2.3) where 1 1 ta~0+ua+-bl(ua)+-2 B2(Ua) v v (van Eeden [13J) -397-

4 NEW APPROXIMATION TO NON-CENTRAL t-distributions 3 B2(Ua)= 9~ {5u;+16u!+3ua+30(4u;+12u;+I) (u!+4ua) (u;-i) ua}. I t is stated in Shibata [12J that the approxin1ation formula (2.3) coincides with that derived from the Taylor expansion of the characteristic function of Tv,a -0 with a chi-square statistic. From the viewpoint of the numerical precision of the approximation formulae (2.1), (2.2) and (2.3), they are not poor when the absolute value of 17: = 0IJ2~+02 is close to 0, but they are poor when 1171 is close to 1, that is, the non-centrality 0 is large (see Table 2). Recently, Akahira [IJ proposed a new higher order approximation formula for a percentage point of the non-central t-distribution and showed that the formula were numerically better than (2.1), (2.2) and (2.3) and also worked well when 1171 is close to 1. Now, by following Akahira [lj, we present a derivation of the new formula. Letting ta be the upper 100a percentage point of the noncentral t-distribution and Z: = X -0, we have l-a=p{tv,o<ta}=p{x/s <ta} =P{(Z +o)/s <ta}=p{z +0 <tas} =P{Z-taS<-o}. Since E[Z-taSJ=tab" and V[Z-taSJ=I+t!(l-b~), we obtain 4) - pi Z-ta(S-bv) tabv-o I ( 2. l-a- Jl+t;(I~"b~) <,ji"+t;(i~b~rj ' and letting (2.5) we easily see that E[WJ=O, V[WJ=I. Note that the statistic TV is based on the chi-statistic S. following theorem. Then we have the THEOREM 2.1 ([IJ). The upper 100a percentile ta of the non-central t-distribution with v degrees of freedom and a non-centrality 0 can be derived fro'ln the formula by ( tabv-o t!(u;-i) 11 1 (" 1)) 2.6),.jl+t;(I-b~)" Ua--24{I+t;(I-b~)}3/2- -v2-+ "4-;?+O --;;4" J. OUTLINE OF THE PROOF. The third and fourth cumulants of T are given and -398-

5 4 J. JAPAN STATIST. SOC. Vol. 25 No respectively. Hence, we have by (2.4), (2.5) and the Cornish-Fisher expan'sion tab)) W ( ( 1 ),J1+t!(1-b~) U a + 6 K3,V[ J u a -1)+T4 K4,v[WJ(ua-3ua)+O 7 =Ua If we ignore the second term of the right-hand side of (2.6), i.e. tabv-o.,j1+t!(1-bn.u a then we have (2.1). Letting bv=::l and 1-b~=::1/(2v), we get (2.2). As previously stated, the van Eeden formula (2.3) is considered to be based on the chi-square statistic. The approximation formula (2.6) is obtained up to the order O(v- 4 ) and derived from the statistic W based on a linear combination of a normal random variable and a chi-statistic S. On the other hand, from Kendall et al. [8J (p. 545) it is well known that a chi-statistic tends to normality with considerably greater rapidity than a chi-square statistic. Hence the approximation formula (2.6) is better on theoretical grounds than that obtained on the basis of the chi-square random variable like (2.3). Fron1 Theorem 2.1 we can obtain the lower confidence limit and the confidence interval for the non-centrality o. COROLLARY 2.1 ([1]). Let T be a statistic according to the non-central t distribution with v degrees of freed01n and a non-centrality parameter o. Then the lower confidence limit 8 of level I-a and the confidence interval [Q, ~J of the non-centrality parameter 0 of level I-a are given by d=bj-u.jl+(l-b;jt'+ 24{i~~~~3)p} (~, + 4~3 )+Op( ~.),,J 2 ( U!/2-1) T3 (1 1) ( 1 ) Q=b"T -U a /2 1+ (1-b v )T2 + 24{1+ (1-b~)T2} 7+ 4v3 +Op 7 ' ~ _,J 2 2 ( U!/2-1) T3 (1 1) ( 1 ) o-bvt +Ua /2 l+(l-b v )T - 24{1+(1-bnT2} 7+ 4v3 +Op 7. The proof is straightforward from Theorem Existence and uniqueness of the solution of the new approximation formula A problem in (2.6) is whether ta exists and is unique or not. tbv-o t 3 (u!-1) ( 1 1) f(t)=fv,a,ii(t):=u a,j1+t2(1-b~) 24{1+t2(1-b~)}3/2 7+ 4v 3 ' o So, denoting we want to solve the equation f(t) =0 by Newton's method. But if the solutions of the equation f(t) = 0 are not unique, there is no guarantee that a solution which is a good approximation to the true value of ta is found by Newton's method. First, we make sure by the graph of f(t). Fig. 1 is the graph of f(t) for a = 0.05, v = 10 and 7J are from 0.9 to 0.1 at intervals of 0.1. Then the solution of the equation f(t) =0 exists uniquely (by Theorem 3.1 this is true). In Fig. 2, for a=0.05, V= 10 and 7J are from -0.9 to -0.1, the solution also exists -399-

6 NEW APPROXIMATION TO NON-CENTRAL t-distributions 5 uniquely. However, this is not generally so. For example, when V= 10 and u a =6 (see Fig. 3 and Fig. 4), from Theorem 3.3, if o~o then the solution does not exist, and if 17:::;; -0.7 then the solutions are not unique. Hence it cannot be said that the solution of (2.6) exists or is unique. In Theorem 3.1, Corollary 3.1, Theorem 3.2 and Corollary 3.2, below show that the solution of the equation f(t) = 0 exists uniquely for practical cases. frt) 40 t Fig.1. Graph of j(t) (v=10, u a = (a=0.05), "1= ) f(t) 40 t Fig. 2. Graph of j(t) (v=10, u a = (a=0.05). "1= ) -400-

7 6 J. JAPAN STATIST. SOC. Vol. 25 No f(t) 't Fig. 3. Graph of f(t) (v=lo, u a =6.0, "7=0.9,...,0.1) f(t) Fig. 4. Graph of f(t) (v=10, u a =6.0, "7=-0.1,...,-0.9) Letting b=b", U=Ua, a=a~:=l-b~ and bt-o g(t)=g",a,a(t) :=u-",j(+-caz-' t 6 y(t)=y~(t) :=-(i-+at 2 )3 (an even function), h(t)=h",(t(t): = ~,a: = sup Ih(t) I, -oo<t<= ( 1 1) ; sgn(t)vy(t) 24 v 2 4v 3 / (an odd function), -401-

8 NEW APPROXIMATION TO NON-CENTRAL t-distributions 7 we havef(t) =g(t)-h(t). LEMMA 3.1. The following assertions hold. O<b,<b,<, b;=l- ;v +O(+'), a,>a,>. 'a,>o, a,= 2 1 v +O( :' ). PROOF. It is obvious that b,,>o. Generally, for s>o r(s) =J27Ts s - C1I2 ) exp l-s+-l--~l (0<8(s) <1) 12s 360s 2 holds. (This is obtainable by letting n = 1 in Whittaker and vvatson [14J (pp ).) Hence we have 2_1( 1)" [ 1 218(V/2) 8((V+l)/2))J b,,--; 1+-; exp - 3v(v+1) + 45 v3 (v+1)3 ' and for v:2:2, we get b~ {I + (l/v))" b~-l [1+{1/(v-1)}],, (v/2). exp [ 3(v+ 1)v(v-1) + 45 v3 :2cexp 13(v+l~V(V-l) 2(13v 2-32v+ 15) =exp 45(v+ 1)v(v-1)3 2 45(V~1)') 8( (v-1 )/2) (v-l )3 8((v+1)/2) 1 J (v+1)3 y' (t) IU;-ll ( 1 1) 1 e : = sup Ih(t) 1= g'(t) 3 --a;;2, -co<t<co V V a" -aot-b (1+at2)3/2' The proof is straightforward. THEOREM 3.1. g(± co):= lim g(t)=u+ :, t-->±co V a (0*0). If l::;;lu a l<b"/ja,,, then the solution of f(t)=o exists -402-

9 8 J. JAPAN STATIST. SOC. Vol. 25 No uniquely. Furthennore, if one of the following (i) 0>0 and {( -b,,/j a" <ua:;' -1 or l:;'ua<bv/j av+ev,a" (ii) 0=0 and l:;'luai<b,,/ja,,+ev,a (iii) 0<0 and {( -blj/j a"-,,,a<ua:;' -1 or l:;'ua<b"/ja,," is satisfied, then the solution of f(t) = 0 exists uniquely. PROOF. 'Vle will prove it assuming (i) and U a*' ± 1. In other cases, we can show it in a similar way. From Lemma 3.2, when 0>0 and lual > 1, the increase and decrease of g, -h and f are given by Table 1. Table l. Increase and decrease of g, -h and f b t ao b b 2 +ao 2 b g(t) u+,j a / u+ '\. u-.; a.; a,j ao 2 +b 2 -h(t) c '\. >0' '\. -c b b b f(t) u+.; a +c >u+.; a '\. u-.; a -c Hence, if u-b/j{i - <O<u+b/Ja, i.e. -bv/j av <ua<blj/j av + lj,a, then the solution of f(t) =0 exists uniquely. D COROLLARY 3.1. We fix a such that luai ~1. For a sufficiently large v which is independent of 0, the solution of f(t) =0 exists uniquely. The proof is straightforward from Theorem 3.1 and Lemlna 3.1. Numerically, we have bl /-2-. /-= Y--2 :::;: , Val 7Tbz ~-7T-. /-= -4-:::;: , va2-7t bs / 8. /-= Y 3 8 =;: , '" as 7Tb4 / 97T. J a4 = Y 32-97T :::;: , and by Lemma 3.1, Theorem 3.1 and a table of the normal distribution, if 0.1:;'a:;'0.15 for v=l, 0.03:;'a:;'0.15 for jj=2, 0.01:;'a:;, :;'a:::;:0.15 for for jj=3, jj~4, then the solution of f(t) = 0 exists uniquely. Also in the case that the inequality above is satisfied for I-a instead of a, the solution exists uniquely. Hence, there is no practical problem when using it for tests. \ivhen we find the -403-

10 NEW APPROXIMATION TO NON-CENTRAL i-distributions 9 median, however, it is impossible to use Theorem 3.1. It is possible to use the following Theorem 3.2, but it is not easy to find out how large of a v we should take. THEOREM 3.2. If the following conditions (C1) to (C4) are satisfied for S01ne y>o, then the solution of f(t) =0 exists uniquely. (C1) Cv,a< ~" -Iual, va" (C2) -g(y) ( = - u.+ );:~"~2) >e". and b"y+o ) g(-y) ( =U + Jl+a"y2>cv,a, a (C3) PROOF. Considering an increase and decrease of 9 from Lemma 3.1, 9 has the unique real zero of order 1 under the condition (C1). We obtain by (C2) that 9 has a real zero on (-y, y), and we have t-::;, -y implies g(t) >, t';::.y implies g(t) < -. Moreover, sincef=g-h and Ih(t)l-::;'c (t E R), we have t-::;,-y implies f(t) >0, t';::.y implies f(t) <0. Hence, f(t) = 0 has at least one solution on (-y, y), and it has no solution outside (-y, y). Next, we extend f, 9 and h to complex functions. We define JZ- so as to be analytic outside {z-::;'o}. Then Jl+az 2 is analytic outside {g(z=o, l~zl';::.l/ja}. We get from (Ca) that 9 and h are analytic in an open set containing {lzl-::;'y}. Further, on Izl =y, we have Ig(z)l;?, ~~I:;I lui, 11 + az 2 1 ';::.1-ay2 ( > 0), y6 ly(z)l-::;' (1-ay2)a ' I u 2-11 ( 1 1) y3 Ih(z)l-::;' 24 \7+ 4v3 (l_ay2)3/2 Since Ih(z)I<lg(z)l on Izi =y from (C4), we get by Rouche's theorem (see e.g. Fisher [3J p. 177) that the number of zeros of f on {izi <y} coincides with that of g. Since the equation g(t) = 0 is essentially quadratic with real moduli and has a real solution on (-y, y), it does not have an imaginary solution and the -404-

11 10 J. JAPAN STATIST. SOC. Vol. 25 No number of solutions of f(t) =0 on {Izl <1'} is one. Since there exists at least one solution on (-1',1'), the solution exists uniquely on (-1', 1')..Moreover, since the real solution does not exist outside (-1', 1'), the real solution of f(t) = 0 exists uniquely. D COROLLARY 3.2. If a and 0 are fixed, then for a sufficiently large v, the solution off(t) = 0 exists uniquely. PROOF. We fix a, 0 and a sufficiently large 1'>0. The assertion follows because (Cl) to (C4) in Theorem 3.2 are satisfied for a large v. D Theorem 3.3 and Corollary 3.3 below show that the solutions of f(t) = 0 may not be unique, and that it may not have a solution. THEOREM 3.3. Let c=c":= ;4 ( :' + 4~8 ) a~/2 D=D":=1-4C"U~" -c,), then the following assertions hold. (1) If the following conditions (Bl) to (B4) are satisfied, then the equation f(t) =0 has at least two solutions. (Bl) (Bs) Dv>O, -1-../75:< < -1+.JR 2 Ua 2 ' Cv c~ (E,) 8>0 and g( - ::8) (=Ua+ ~a"b~:~8~:b~ ) 20. (2) If (Bl) to (Bs) in (1) and 0~0 are satisfied, then f(t) =0 does not have a solution. (3) We may replace (a, 0) by (I-a, -0) in the assumptions of (1) and (2), respectively. PROOF. (1) From (Bl) to (Bs), we have u<-l. From this and the definition of c, we can see that = (u 2-1)c. Hence we have f(-oo)=u+ )a +(U'-1)C=CU 2 +u+()a -C). Considering that f( - 00) is a quadratic form of u, we have f( - 00) <0 from (Bl) and (B2). From (B4) and u< -1, we have f( -~) >g(-~ ) ~O, \ ao ao / and from u<-i, we have b f(oo)<u-../ a - <u< -1<0. Hence, the equation f(t) = 0 has at least two solutions. ' -405-

12 NEW APPROXIMATION TO NON-CENTRAL t-distributions 11 (2) Considering an increase and decrease ofjby 0~0, we seej(t) <j(-oo), and from (Bl) to (B3), we havej(-oo)<o. Hence we can seej(t) <0. (3) This is obtained by j",l-a,-a(-t) = -j",a,a(t). 0 COROLLARY 3.3. Ij a sufficiently large v is fixed, then there exists (a, 0) such that the solutions oj the equation j(t) = 0 are not unique, and there also exists (ex, 0) such that it does not have a solution. PROOF. In Theorem 3.3, (Bl) and (B3) are conditions with respect to v. From Lemma 3.1, we have 1-1+.JD: lid =-00 lj-+co 2c" ' hence (Bl) and (B3) are satisfied for a sufficiently large v. For such v, we fix ex satisfying (Bz). Since limhoo g",a,a(-bjao) = 00, (B4) is satisfied for a sufficiently large O. Then the solutions of j(t) = 0 are not unique. For v above, ex above, and 0~0, j(t) =0 has no solution. 0 REMARK 3.1. The reader might think that there is a contradiction because Corollary 3.1 and Corollary 3.2 assert that the solution exists uniquely when v is sufficiently large while Corollary 3.3 asserts that it fails when v is sufficiently large. Because the orders of fixing a, 0 and v are different, however, there is no contradiction. In Corollary 3.1, we fix IX first and fix v later. In Corollary 3.2, we fix a and 0, and fix v later, but in Corollary 3.3, first we fix v, and fix ex and 0 later. In Corollary 3.1, we cannot take v which is independent of not only 0 but also a, because lual > 1 in Theorem 3.3. REMARK 3.2. An algebraic approach to an examination on a solution of the equation j(t) =0 suggested by the referee is as follows. The equation (3.1) implies (3.2) bt-o j(t) =u.j1 +atz If we obtain all the real solutions of the algebraic equation (3.2), this is sufficient in order to ascertain whether or not any of them can serve as a solution of (3.1). In this case, it is possible to obtain numerical solutions of (3.2) by using various methods, e.g. a construction of a Sturm sequence of (3.2). 4. Evaluation of the new approximation formula in comparison with others by numerical calculation In order to compare (2.6) with (2.1), (2.2) and (2.3), we have their numerical calculation when ex is 0.10, 0.05 and 0.01, v is 4, 9, 16, 36. (A similar table on (2.1), (2.2) and (2.3) for a=0.05 is given by Shibata [12].) The errors of the approximation formulae are given as Table 2, where the true values are referred from Yamauti [15J and the values of (2.6) are calculated by Newton's method in Mathematica for Macintosh. As is seen in Table 2, the approximation formula (2.6) dominates the others where the absolute value of -406-

13 12 J. J AP AN STATIST. SOc. Vol. 25 No Table 2. Errors of the approximation formulae of ta for 17=oj.j2v+o 2 (0:.=0.10) v 17 I true value I value (2.6) I error (2.6) I error (2.1) I error (2.2) I error (2.3) O.Oll

14 NEW APPROXIMATION TO NON-CENTRAL i-distributions 13 Table 2. (Continued.) (x=0.05) lj I true value I value (2.6) I error (2.6) I error (2.1) I error (2.2) I error (2.3)

15 14 J. J AP AN STATIST. SOC. Vol. 25 No Table 2. (Continued.) (a=o.oi) lj I true value I value (2.6) I error (2.6) I error (2.1) I error (2.2) ) error (2.3) ;

16 NEW APPROXIMATION TO NON-CENTRAL t-distributions 15 Table 3. Values of to.oz5 for v=nl +nz-2 and o=vnln2/(nl +nz)d. (D=2.0) n1\n ' n1\n

17 16 J. JAPAN STATIST. SOC. Vol. 25 No Table 3. (Continued.) (D=l.O) ) nl\n nl\nz

18 NEW APPROXIMATION TO NON-CENTRAL i-distributions 17 Table 3. (Continued.) (D=0.5) I I I I I I I I I I I I I nl\n nl \n ;

19 18 J. JAPAN STATIST. SOC. Vol. 25 No n is close to 1 and gives precise values on the whole. For example, if a = 0.05, v is more than 9, and Inl is less than 0.70, then the absolute values of errors are less than The values of the upper percentiles of a non-central t-distribution for the t-statistic are given using the approximation formula (2.6) in Table 3. In cases where a=0.025, D is 2.0, l.0 and 0.5, and nl, n2 are from 5 to 30 at unit interval. If D is 2.0, then Inl is less than 0.621, hence the values in Table 3 are reliable. Table 3 may be useful for the argument between statistical significant difference and clinical significant difference (see Kubo et al. [9J). Acknowledgment The authors wish to thank the referee for suggesting Remark 3.2 on the algebraic approach. REFERENCES [ 1 J Akahira, M. (1993). A higher order approximation to a percentage point of the non-central t-distribution. Mathematical Research Note , Institute of Mathematics, University of Tsukuba. To appear in Commun. Statist.-Simula., 24(3). [ 2 J Bagui, S. C. (1993). CRC Handbook of Percentiles of Non-Central t-distributions. CRC Press, Florida. [3 J Fisher, S. D. (1990). Complex Variables (2nd Ed.). Wadsworth & Brooks/Cole, California. [4 J Institute of Mathematical Statistics (ed.) (1974). Selected Tables in Mathematical Statistics Vol. 2. American Mathematical Society, Providence. [5 J Jennett, W. J. and Welch, B. L. (1939). The control of proportion defective as judged by a single quality characteristic varying on a continuous scale. J. Roy. Statist. Soc. Suppl., 6, [ 6 ] Johnson, N. L. and Kotz, S. (1970). Continuous Univariate Distributions, 2. Wiley, New York. [ 7 J Johnson, N. L. and Welch, B. L. (1940). Applications of the non-central t-distribution. Biometrika, 31, [8 J Kendall, Sir M., Stuart, A. and Ord, K. (1994). Kendall's Advanced Theory of Statistics Vol. 1 (6th Ed.). Edward Arnold, London. [ 9 ] Kubo, T., Shigemitsu, S., Inaba, J. and Kasagi, K. (1991). Statistical significant difference and clinical significant difference. (In Japanese). 11th] oint Congress of Medical Informatics, [10J Masuyama, M. (1951). An approximation to non-central t-distribution with the stochastic paper. Rep. Stat. Appl. Res., JUSE, 1, [l1j Owen, D. B. (1968). A survey of properties and applications of the non-central t distributions. Technometrics., 10, [12J Shibata, Y. (1981). Normal Distributions. (In Japanese). Tokyo Univ. Press, Tokyo. [13J van Eeden, C. (1961). Some approximations to the percentage points of the noncentral t-distribution. Int. Statist. Rev., 29, [14J Whittaker, E. T. and Watson, G. N. (1927). Modern Analysis (4th Ed.), Cambridge Univ. Press, London. [15J Yamauti, Z. et al. (1972). Statistical Tables. (In Japanese). JSA, Tokyo